Problem 32 - A Graphical Approach to Precalculus with Limits

Question

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\sec \frac{\pi}{3}$$

Step-by-Step Solution

Answer

The exact value is 2.

Step 1: Recognize the trigonometric identity

The secant function, \(\sec(\theta)\), is the reciprocal of the cosine function. Thus, \(\sec(\theta) = \frac{1}{\cos(\theta)}\).

Step 2: Determine the cosine value

For \(\theta = \frac{\pi}{3}\), which is equal to 60 degrees, the cosine value is \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\).

Step 3: Calculate the secant value

Using the identity from Step 1, \(\sec\left(\frac{\pi}{3}\right) = \frac{1}{\cos\left(\frac{\pi}{3}\right)} = \frac{1}{\frac{1}{2}} = 2\).

Step 4: Provide decimal approximation if needed

Since the exact value \(2\) is not irrational, a decimal approximation is not required.

Key Concepts

Secant Function

The secant function, denoted as \( \sec(\theta) \), is a key trigonometric function that often pops up in mathematical problems. It's closely related to the cosine function, which you might be more familiar with. In fact, secant can be understood as the reciprocal of the cosine function. This means that for any angle \( \theta \), the secant value is given by \( \sec(\theta) = \frac{1}{\cos(\theta)} \). By understanding this relationship, many problems involving trigonometric identities can be simplified.

In practical terms, if you know the cosine of an angle, you can easily find the secant by taking the reciprocal (or 'flipping' the fraction). For example, if \( \cos(\theta) = \frac{1}{2} \), then \( \sec(\theta) = 2 \). Understanding the secant function in this way will help you solve problems that require you to find unknown angles or evaluate expressions involving trigonometric functions.

Trigonometric Identities

Trigonometric identities are the essential building blocks in the world of trigonometry. These identities are equations that are true for all values of the variable, usually represented as angles. Understanding these identities helps in simplifying trigonometric expressions and solving equations.

One crucial identity to remember is the relationship between sine and cosine, expressed through the Pythagorean identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \). Another key identity is the reciprocal identity, which relates the secant and cosine functions: \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Knowing these identities, you can break down complex problems into simpler forms.

• Reciprocal Identities: These identities link trigonometric functions with their reciprocals. For example, \( \sec(\theta) \) connects directly to \( \cos(\theta) \).
• Pythagorean Identities: These extend the fundamental relationship between \( \sin \) and \( \cos \) functions.
• Quotient Identities: These show the relationship among \( \tan(\theta), \sin(\theta), \) and \( \cos(\theta) \).

Knowing these identities not only helps with solving equations but also enhances your understanding of trigonometry as a whole.

Cosine Value

The cosine value of an angle is an essential concept in trigonometry. Cosine is a fundamental trigonometric function that measures the ratio of the adjacent side to the hypotenuse in a right-angled triangle. It's often used to determine angles and lengths in various applications, from architecture to physics.

For specific angles, like \( \frac{\pi}{3} \) (or 60 degrees), the cosine value is a well-known constant: \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \). This information is crucial, especially when dealing with problems involving the secant function, as you can easily find the secant by taking the reciprocal of the cosine. When solving trigonometric problems, it is helpful to memorize cosine values for common angles such as 0, \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), and \( \frac{\pi}{2} \).

• Notable Angle Values: Memorizing values for notable angles makes computations faster.
• Application in Triangles: Used in cosine rules and calculating distances.

By understanding these basic cosine values and how they are utilized with other trig functions like secant, you can tackle a wide range of math problems more efficiently.